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Frédéric Bayart

Publications and source records attributed to Frédéric Bayart.

47 records · Page 3Linked to original sources
arXiv2013

Hypercyclic operators and rotated orbits with polynomial phases

An important result of León-Saavedra and Müller says that the rotations of hypercyclic operators remain hypercyclic. We provide extensions of this result for orbits of operators which are rotated by unimodular complex numbers with polynomial phases. On the other hand, we show that this fails for unimodular complex numbers whose phases grow to infinity too quickly, say at a geometric rate. A further consequence of our work is a notable strengthening of a result due to Shkarin which concerns variants of León-Saavedra and Müller's result in a non-linear setting.

math.FA↗
arXiv2012

Multifractal spectra of typical and prevalent measures

We prove that, in the Baire category sense, a typical measure supported by a compact set admits a linear lower singularity spectrum. We investigate the same question for the upper singularity spectrum and for other forms of genericity.

math.PR↗
arXiv2012

The multifractal box dimensions of typical measures

We compute the typical (in the sense of Baire's category theorem) multifractal box dimensions of measures on a compact subset of $\mathbb R^d$. Our results are new even in the context of box dimensions of measures.

math.CA↗
arXiv2012

Boundary multifractal behaviour for harmonic functions in the ball

It is well known that if $h$ is a nonnegative harmonic function in the ball of $\RR^{d+1}$ or if $h$ is harmonic in the ball with integrable boundary values, then the radial limit of $h$ exists at almost every point of the boundary. In this paper, we are interested in the exceptional set of points of divergence and in the speed of divergence at these points. In particular, we prove that for generic harmonic functions and for any $β\in [0,d]$, the Hausdorff dimension of the set of points $ξ$ on the sphere such that $h(rξ)$ looks like $(1-r)^{-β}$ is equal to $d-β$.

math.CA↗
arXiv2012

How behave the typical $L^q$-dimensions of measures?

We compute, for a compact set $K\subset\mathbb R^d$, the value of the upper and of the lower $L^q$-dimension of a typical probability measure with support contained in $K$, for any $q\in\mathbb R$. Different definitions of the "dimension" of $K$ are involved to compute these values, following $q\in\mathbb R$.

math.CA↗
arXiv2011

On the Hausdorff dimension of graphs of prevalent continuous functions on compact sets

Let $K$ be a compact set in $\rd$ with positive Hausdorff dimension. Using a Fractional Brownian Motion, we prove that in a prevalent set of continuous functions on $K$, the Hausdorff dimension of the graph is equal to $\dim_{\mathcal H}(K)+1$. This is the largest possible value. This result generalizes a previous work due to J.M. Fraser and J.T. Hyde which was exposed in the conference {\it Fractal and Related Fields~2}. The case of $α$-Hölderian functions is also discussed.

math.CA↗
arXiv2011

Mixing operators and small subsets of the circle

We provide complete characterizations, on Banach spaces with cotype 2, of those linear operators which happen to be weakly mixing or strongly mixing transformations with respect to some nondegenerate Gaussian measure. These characterizations involve two families of small subsets of the circle: the countable sets, and the so-called sets of uniqueness for Fourier-Stieltjes series. The most interesting part, i.e. the sufficient conditions for weak and strong mixing, is valid on an arbitrary (complex, separable) Fréchet space.

math.FA↗
arXiv2011

Multifractal analysis of the divergence of Fourier series

A famous theorem of Carleson says that, given any function $f\in L^p(\TT)$, $p\in(1,+\infty)$, its Fourier series $(S_nf(x))$ converges for almost every $x\in \mathbb T$. Beside this property, the series may diverge at some point, without exceeding $O(n^{1/p})$. We define the divergence index at $x$ as the infimum of the positive real numbers $β$ such that $S_nf(x)=O(n^β)$ and we are interested in the size of the exceptional sets $E_β$, namely the sets of $x\in\mathbb T$ with divergence index equal to $β$. We show that quasi-all functions in $L^p(\TT)$ have a multifractal behavior with respect to this definition. Precisely, for quasi-all functions in $L^p(\mathbb T)$, for all $β\in[0,1/p]$, $E_β$ has Hausdorff dimension equal to $1-βp$. We also investigate the same problem in $\mathcal C(\mathbb T)$, replacing polynomial divergence by logarithmic divergence. In this context, the results that we get on the size of the exceptional sets are rather surprizing.

math.CA↗
arXiv2009

Composition operators on the Wiener-Dirichlet algebra

We study the composition operators on an algebra of Dirichlet series, the analogue of the Wiener algebra of absolutely convergent Taylor series, which we call the Wiener-Dirichlet algebra. The central issue is to understand the connection between the properties of the operator and of its symbol, with special emphasis on the compact, automorphic, or isometric character of this operator. We are led to the intermediate study of algebras of functions of several, or countably many, complex variables.

math.FA↗
arXiv2004

Composition operators on the Wiener-Dirichlet algebra

In this paper, we study the composition operators on an algebra of Dirichlet series, the analogue of the Wiener algebra of absolutely convergent Taylor series, which we call the Wiener-Dirichlet algebra. We study the connection between the properties of the operator and of its symbol, with special emphasis on the compact, automorphic, or isometric character of this operator. We are led to the intermediate study of algebras of functions of several, or countably many, complex variables.

math.FA↗